Upload
others
View
5
Download
0
Embed Size (px)
Citation preview
Lecture2:GPSPseudorange
GEOS655TectonicGeodesyJeffFreymueller
GPSDesignTimeline• NAVSTAR=NAVigaEonSystem
withTimingandRanging• Always-on,instantglobal
posiEoning• Developmentbeganin1973• Firstsatelliteslaunched1978• Userequiptests1980• 1983:KoreanAir007shotdownby
SovietUnion– Planehadstrayedaconsiderable
distanceintoSovietairspace– LedUSPres.Reagantomandate
futurecivilianuseofGPS
GPSUserHardware–Old!
GPSUserHardware–Modern!
DifferentModesofUse
Naviga&on• Instantaneous• SinglestaEon• Originalintendeduse• Accuracy
– Fewmeters– Sub-meterw/differenEalcorrecEons
Surveying• Usuallypost-process• UsuallymulE-staEon• Scienceorsurvey• Accuracy
– 1-2cmatworst– 1-2mmatbest
• Also“seismology”
BasicPrinciples:Surveying• Requiresdatafromn≥4satellites,m≥2receivers
– PointposiEoningapproachesworkwith1receiver• RequiresconEnuoustrackingoverEme• Post-processedbutreal-Emebeingdeveloped• Usepseudorangeandcarrierphasemeasurementsfromeachsatellitetoreceiver
• OrbitsofsatellitesfixedoresEmated• ClockerroronsatellitesesEmatedordifferencedout• EsEmatereceiverposiEon(X,Y,Z)andclockerror• Modelawidevarietyofpathdelaysandothereffects
PosiEoningByRanging1
A 2D example: If you know you are a certain distance from Boise, your position could be anywhere on the circle.
PosiEoningByRanging2
With two distances, you know you are at one of two points.
PosiEoningByRanging3
With three distances, you know you are in Denver. In 3D the circles become spheres, and with three distances you still
have two possible locations – one on the surface and one out in space.
GPSPosiEoning• MeasureposiEonbymeasuringrangestosatellites– Afewsatellitescanserveanunlimitednumberofusersontheground,anywhereintheworld
• Howdoweknowwheresatellitesare?– TheybroadcasttheirposiEons(orbits)inanaviga&onmessage
– (or)someonegivesuspreciseorbits
• Measuredrangesarecalledpseudoranges• High-precisionGPSusesthephaseoftheGPScarriersignaltomeasurechangesinrange
Whycallita“pseudorange”?
• Rangeisthedistancefromsatellitetoreceiver,pluspathdelays.
• Pseudorangeisdistancepluseffectsofclockerrors
• TheterminologyhasoldrootsinnavigaEon.– VLBIandGPSarepseudoranging;SLRisranging
• Geometricrangeρistruedistance.• P=ρ+c*(clockerrors)+c*(pathdelays)
EvolvingSatellites
1970s-1980s: smaller, relatively simple 2000s: big, more complex
SatelliteConstellaEon
SatelliteConstellaEonFacts• Nominally4satellites(SVs)ineachof6equallyspacedorbitalplanes(now5ineachplane).
• Orbitalplanesinclined55°fromequator.• NearlycircularorbitsR=26,600km~4RE• Orbitalperiodis11h58m,twoorbitspersiderealday
• Siderealdayislengthofdaydefinedbywhenstarsappearinsameplaceinsky– DiffersfromrotaEonaldaybecauseofmoEonofeartharoundthesun.
Orbits
• CanesEmateorbitsorfixorbitstopre-determinedvalues• RepresentaEonoforbit
– Broadcast:Keplerianelements+Eme-dependentcorrecEons– TabularfileofXYZsatelliteposiEons– Trajectory:iniEalcondiEons+integrateequaEonsofmoEon
(neededtoesEmateorbits)
• InpracEce,highlypreciseorbitsareavailablefromtheIGS– Ultra-Rapid:includespredict-aheadforrealEmeuse– Rapid:Availablenextday– Final:Availablein<2weeks
KeplerianElements• AnellipEcalorbitandtheposiEonofabodycanbedescribedby6parameters(Keplerianelements)– Semi-majoraxis– Eccentricity– Fouranglesshownatles
• Earthisnotapointmass,sosatelliteorbitsarenotexactlyellipEcal.– Otherforcesalsoperturborbit
SatelliteGroundTracks
24hoursofGPSDataskytracks
Southern California Fairbanks
These are the paths you would see in the sky if you could see the satellites
GPSSignalStructure• ThreefrequenciesatL-band,L1,L2,andL5
– L1at154*10.23MHz(~19cm)– L2at120*10.23MHz(~24cm)– L5at115*10.23MHz(~25cm)
• CodesModulated(phasemodulaEon)ontoeachcarrier– P-codeat10.23MHzonL1+L2– C/A(CoarseAcquisiEon)codeat1.023MHzonL1+L2(newL2C)
– NavigaEonmessageat50bitspersecond
• PandC/Acodesaretypesofpseudo-randomnoise(PRN)codes
TypesofsignalmodulaEon
Amplitude Modulation (AM)
Frequency Modulation (FM)
Phase Modulation (PM)
SatelliteSignals
PrecisionofObservaEons
• Code“chiplength”isthedistanceassociatedwitheachbitofthecode.– C/A:293m
• Repeatsevery~300km
– P:29.3m
• Carrierwavelengthisanalogoustochiplength==2-3ordersofmagnitudemoreprecise
“chip length”
DenialofAccuracy• USDoDcanreduceaccuracyforreal-Emecivilianusers• (S/A)SelecEveAvailability–onfrom1990stolate1990s– Epsilon(introduceerrorsinnavigaEonmessage)– Dither(introducerapidvariaEoninSVclocks)– Militaryreceivershavespecialchipstoundothis
• (A/S)AnE-Spoofing–onsince1994– EncrypEonofP-code– Prevents“theenemy”fromimitaEng(spoofing)GPSsignal– ModernreceiversgetaroundthisencrypEoninvariousways(includinganear-reverse-engineeringinonecase).
Pseudo-RandomNoise• Computerscannotgeneratetruerandomnumbers,butcangenerateasequenceofnumberswithrandomstaEsEcalproperEes.– Butthesequencecanberepeatedexactly– BeginwithsomestarEngvalue,thenperformaseriesofoperaEons
• C/Acodehas1023bits,repeats1000Emespersecond
• Pcodehasalotofbits,repeatsevery266.4days;eachSVgetsa7-daypieceofcode
CodeCorrelaEonforRanging
Receiver generates a copy of the (known) code and correlates with the received code
PseudorangeObservaEonModel• ThecorrelaEonEmeshisgivesanesEmateofthetravelEme,whichisthefundamentalpseudorangemeasurement.– TravelEme=(EmeofrecepEon)–(Emeoftransmission)
• PS=(T–TS)c– T=receiverclockreadingatrecepEon– TS=satelliteclockreadingattransmission– c=speedoflight=299792458m/s
AccounEngforClockBiases• Clockbiasorclockerror?
– Error==mistake– Error==bias,measurementerror– Error==esEmateofuncertaintyintheabove
• VLBI“removed”clockerrorsbyusingultra-stablehydrogenmaserclocks– VLBIactuallymodelsclockbiasesasquadraEc
• GPSmustesEmatereceiverclockbias(andsatelliteclockbiasforhighprecisionwork)
ObservaEonModelwithClocks• PS=(T–TS)c
– T=t+τ– TS=tS+τS– t,tSaretruereceive,transmitEmes,τareclockerrors
• SubsEtuEng– PS=[(t+τ)–(tS+τS)]c– PS=(t-tS)c+(τ–τS)c– PS=ρS(t,tS)+(τ–τS)c
• ρS(t,tS)israngefromreceiveratreceiveEmetosatelliteattransmitEme:
€
ρS (t, t S ) = xS (t S ) − x(t)( )2
+ yS (t S ) − y(t)( )2
+ zS (t S ) − z(t)( )2
|τ|≤1millisecond
|τS|issmall(CesiumorRubidiumclocks)
LightTimeEquaEon• ThetransmissionEmeis~0.07sec.ToevaluatethegeometricrangeweneedtomapthesatelliteposiEonbacktothetransmissionEme.ButwestartoutknowingonlythereceiveEme.WecansolvethisproblemiteraEvely:
€
t(0)S = t = (T − τ)
t(1)S = t −
ρS (t,t(0)S )
c
t(2)S = t −
ρS (t,t(1)S )
c!
First guess: Transmit time = receive time
Next iteration: Correct for satellite position based on the transmit time estimated from the previous iteration.
SetofSimplifiedObserv.EquaEons
• Now,generalizetomulEplesatellites.WeuseasuperscripttoidenEfyeachsatellite(don’tconfusewithanexponent).LaterwewillhavetouseasubscripttokeeptrackofmulEplereceivers:– P(1)=[(x(1)–x)2+(y(1)–y)2+(z(1)–z)2]1/2+cτ–cτ(1)– P(2)=[(x(2)–x)2+(y(2)–y)2+(z(2)–z)2]1/2+cτ–cτ(2)– P(3)=[(x(3)–x)2+(y(3)–y)2+(z(3)–z)2]1/2+cτ–cτ(3)– P(4)=[(x(4)–x)2+(y(3)–y)2+(z(3)–z)2]1/2+cτ–cτ(4)
x, y, z, τ: receiver position and clock error
LinearizingNonlinearEquaEons• Therearesimplewaysto
solvesystemsoflinearequaEons,likematrixinversionorleastsquares.Butwehaveanonlinearproblem.Oneapproachistolinearize,orconstructalinearapproximaEontothenon-linearproblem.WecandothatwithTaylor’stheorem(TaylorSeries)
f(x)=
LinearizingPart2
• WeapproximatebytakingjustthelineartermsoftheTaylorSeries.
• Welinearizeaboutapproximatevalues(a,b)
• ParEalderivaEvesarecomputedat(a,b)
LinearizingOurEquaEons
• WelinearizeourequaEonsaboutapproximatevalues(x0,y0,z0,τ0)
€
P(x,y,z,τ ) = P(x0,y0,z0,τ 0) +∂P∂x
x − x0( ) +∂P∂y
y − y0( ) +∂P∂z
z − z0( ) +∂P∂τ
τ − τ 0( )
P(x,y,z,τ ) = P(x0,y0,z0,τ 0) +∂P∂x
Δx +∂P∂y
Δy +∂P∂zΔz +
∂P∂τ
Δτ
P(x,y,z,τ ) − P(x0,y0,z0,τ 0) =∂P∂x
Δx +∂P∂y
Δy +∂P∂zΔz +
∂P∂τ
Δτ
Pobserved − Pcomputed =∂P∂x
Δx +∂P∂y
Δy +∂P∂zΔz +
∂P∂τ
Δτ
ObservaEonModel• TosolvetheseequaEons,weneedtowritethemintheformofanobservaEonmodel.– ObservaEons=Model+measurementnoise
• Pobserved=P(x,y,z,τ)+v
€
Pobserved = Pcomputed +∂P∂x
Δx +∂P∂y
Δy +∂P∂zΔz +
∂P∂τ
Δτ + v
Pobserved − Pcomputed =∂P∂x
Δx +∂P∂y
Δy +∂P∂zΔz +
∂P∂τ
Δτ + v
ΔP =∂P∂x
∂P∂y
∂P∂z
∂P∂τ
⎛
⎝ ⎜
⎞
⎠ ⎟
ΔxΔyΔzΔτ
⎛
⎝
⎜ ⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟ ⎟
+ v
MatrixEquaEon
• ItiseasiertodealwiththisequaEonifwewriteitasamatrixequaEon:
€
ΔP (1)
ΔP (2)
ΔP (3)
ΔP (4 )
⎛
⎝
⎜ ⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟ ⎟
=
∂P(1)
∂x∂P(1)
∂y∂P(1)
∂z∂P(1)
∂τ∂P(2)
∂x∂P(2)
∂y∂P(2)
∂z∂P(2)
∂τ∂P(3)
∂x∂P(3)
∂y∂P(3)
∂z∂P(3)
∂τ∂P(4 )
∂x∂P(4 )
∂y∂P(4 )
∂z∂P(4 )
∂τ
⎛
⎝
⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟
ΔxΔyΔzΔτ
⎛
⎝
⎜ ⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟ ⎟
+
v(1)
v(2)
v(3)
v(4 )
⎛
⎝
⎜ ⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟ ⎟
EvaluatetheParEalDerivaEves
• Thisisosenwriyeninmatrixformlike– b=Ax+v Aiscalledthe“Designmatrix”– Ifρ(i)=[(x0–x(i))2+(y0–y(i))2+(z0–z(i))2]1/2
€
A =
x0 − x(1)
ρ(1)y0 − y
(1)
ρ(1)z0 − z
(1)
ρ(1)c
x0 − x(2)
ρ(2)y0 − y
(2)
ρ(2)z0 − z
(2)
ρ(2)c
x0 − x(3)
ρ(3)y0 − y
(3)
ρ(3)z0 − z
(3)
ρ(3)c
x0 − x(4 )
ρ(4 )y0 − y
(4 )
ρ(4 )z0 − z
(4 )
ρ(4 )c
⎛
⎝
⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟
These have the form of trig functions, and can also be written in terms of the azimuth to the satellite and
the inclination of the satellite above the
horizon.
SolvingtheEquaEons
• Ifthereare4observaEonsexactly,thenthesystemofequaEonscanbesolvedexactly:– b=Ax+v èx=A-1b(noisev=0isassumed)
• Ingeneral,wewillhavemorethan4satellitesobservedataEme.Sohowdowefindthe“best”soluEon.Leastsquares!– LeastsquaressoluEonminimizesthesumofsquaresofresiduals,thatis
– FindthexthatgivestheminimumvTv.
LeastSquaresSoluEon
• TheleastsquaressoluEon,forequallyweighteddata,is– x’=(ATA)-1ATb
• Thisassumesthat(ATA)-1exists.Itwillexistaslongasthereare4ormoresatelliteslocatedindisEnctdirecEonsinthesky.Twosatelliteslocatedinexactlythesameplacewouldcountasone.
BiasesandErrors• Supposeweknowwhatthe
measurementerrorsare.HowwouldtheseknownerrorsbiasouresEmatesofposiEon(andclockbias)?
– vx=(ATA)-1ATv– Youmightusethistodetermine
whetheranewlydiscoverederrorhasabigimpactonyouresEmatedparameters,butingeneralyouwouldsimplywanttocorrectthedata!
• Inreality,youdon’tknowthemeasurementerrors,butyoumayknowtheirstaEsEcalproperEes.Forexample,youmayknowthatthemeanmeasurementerroris0,withuncertaintyσ,andthatthemeasurementerrorsfollowaGaussianornormaldistribu&on.
In this case: Expectation: E(v) = 0 Covariance: C = E(vvT) = σ2I
CovarianceMatrix• Thecovarianceisgivenasa(symmetric)matrix.Fortheproblemwehavejustsolved,– Cx=σ2(ATA)-1
– σ2isthedatanoise.(ATA)-1relatesonlytothegeometry.
• Orintermsofthecomponents:
€
Cx =σ 2
σ x2 σ xy σ xz σ xτ
σ yx σ y2 σ yz σ yτ
σ zx σ zy σ z2 σ zτ
στx στy στz στ2
⎛
⎝
⎜ ⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟ ⎟
LocalCoordinates
• YoucanalsotransformtheXYZcoordinatestolocalcoordinates(east,north,height).We’llleavetheequaEonsforthatforlater.Butifyoutakejustthecoordinatespartofthecovariance,youget:
€
CL =σ 2
σ e2 σ en σ eh
σ ne σ n2 σ nh
σ he σ hn σ h2
⎛
⎝
⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟
DOPs–DiluEonofPrecision• YourhandheldGPSprobablyreportsanumbercalled“PDOP”,whichstandsfor“PosiEonDiluEonofPrecision”.TheseareotherDOPsaswell,whichallgivemeasuresofhowthesatellitegeometrymapsintoposiEonorEmeprecision.– VDOP=σh
– HDOP=(σe2 + σn
2)1/2 – PDOP=(σe
2 + σn2 + σh
2)1/2 – GDOP=(σe
2 + σn2 + σh
2 + c2στ2)1/2
– TDOP=στ• MulEplyPDOPbymeasurementprecisiontogetuncertaintyin3DposiEon.
PDOP > 5 considered poor